Derivative of \( \displaystyle - \frac{\ln{\left(\sin{\left(2 x \right)} - 1 \right)}}{4} + \frac{\ln{\left(\sin{\left(2 x \right)} + 1 \right)}}{4} \)

Problem 2.60 · hard Beautiful

1. \[ \frac{d}{d x} \left(- \frac{\ln{\left(\sin{\left(2 x \right)} - 1 \right)}}{4} + \frac{\ln{\left(\sin{\left(2 x \right)} + 1 \right)}}{4}\right) \]
   derivativeStart with the derivative of the function.✓ Proved
2. \[ = \frac{d}{d x} \left(- \frac{\ln{\left(\sin{\left(2 x \right)} - 1 \right)}}{4}\right) + \frac{d}{d x} \frac{\ln{\left(\sin{\left(2 x \right)} + 1 \right)}}{4} \]
   sum constant-multipleApply the sum rule. Factor out the constant coefficients.✓ Proved
3. \[ = - \frac{\frac{d}{d x} \ln{\left(\sin{\left(2 x \right)} - 1 \right)}}{4} + \frac{\frac{d}{d x} \ln{\left(\sin{\left(2 x \right)} + 1 \right)}}{4} \]
   constant-multipleDistribute the constant factor.✓ Proved
4. \[ = \frac{\frac{d}{d x} \left(\sin{\left(2 x \right)} + 1\right)}{4 \left(\sin{\left(2 x \right)} + 1\right)} - \frac{\frac{d}{d x} \left(\sin{\left(2 x \right)} - 1\right)}{4 \left(\sin{\left(2 x \right)} - 1\right)} \]
   chainApply the chain rule to the logarithmic terms.✓ Proved
5. \[ = \frac{\cos{\left(2 x \right)} \frac{d}{d x} 2 x}{4 \left(\sin{\left(2 x \right)} + 1\right)} - \frac{\cos{\left(2 x \right)} \frac{d}{d x} 2 x}{4 \left(\sin{\left(2 x \right)} - 1\right)} \]
   chainApply the chain rule to the sine terms.✓ Proved
6. \[ = \frac{\cos{\left(2 x \right)}}{2 \left(\sin{\left(2 x \right)} + 1\right)} - \frac{\cos{\left(2 x \right)}}{2 \left(\sin{\left(2 x \right)} - 1\right)} \]
   derivative algebraDifferentiate the inner function 2*x. Simplify the coefficients and fractions.✓ Proved
7. \[ = \frac{\left(\frac{1}{\sin{\left(2 x \right)} + 1} - \frac{1}{\sin{\left(2 x \right)} - 1}\right) \cos{\left(2 x \right)}}{2} \]
   algebraFactor out the common term 1/2 * cos(2*x).✓ Proved
8. \[ = - \frac{\cos{\left(2 x \right)}}{\left(\sin{\left(2 x \right)} - 1\right) \left(\sin{\left(2 x \right)} + 1\right)} \]
   algebraFind a common denominator for the terms in the parentheses.✓ Proved
9. \[ = - \frac{\cos{\left(2 x \right)}}{\sin^{2}{\left(2 x \right)} - 1} \]
   algebra simplify algebraExpand the numerator and denominator. Combine like terms in the numerator. Simplify the fraction by canceling the 2.✓ Proved
10. \[ = \frac{\cos{\left(2 x \right)}}{1 - \sin^{2}{\left(2 x \right)}} \]
    algebraMultiply the numerator and denominator by -1.✓ Proved
11. \[ = \frac{1}{\cos{\left(2 x \right)}} \]
    rewrite simplifyUse the trigonometric identity 1 - sin^2(u) = cos^2(u). Simplify the fraction by canceling one cos(2*x).✓ Proved
12. \[ = \sec{\left(2 x \right)} \]
    rewriteUse the definition of the secant function.✓ Proved